the economics of rotating savings and credit associations by timothy besley, stephen coate and glenn...
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The Economics of Rotating Savings and Credit Associations
By Timothy Besley, Stephen Coate and Glenn Loury
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What is a ROSCA?
Rotating Savings and Credit Associations (Roscas) are an
informal financial institutions where a group of individuals
come together to make regular contributions to a common
fund, which is then lent out in whole to one member each cycle
– giving that member access to a larger amount of capital than
would otherwise be possible.
It is a poor man’s bank!
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Where would you find it?
• Roscas are found all over the world mainly in developing
countries namely India (chit funds), Senegal (tontines),
Cameroon (njangis), Sri Lanka (cheetu) etc
• It is also used by immigrant groups in United States.
• Many US savings and loan associations started life as
Roscas
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Motivation behind Roscas
• In the absence of external funds they can be used to finance
lumpy expenditures.
• They play a role in transferring resources to meet life-cycle
needs.
• They provide a means of making joint savings work.
• Roscas can be understood as a response by a socially
connected group to credit market exclusion.
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Types of Roscas• Random Roscas:
- members commit to putting a fixed amount into a pot for each period of life of the Rosca
- pot is allocated randomly to one of the members
- process continues next period with the winner of the period excluded until everyone has been covered
• Bidding Roscas:
- similar to Random Roscas
- winner is determined by a bidding procedure
- Bids are in form of pledge of higher contributions
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Problems with Roscas
• Not useful for buffering against risk
- In Random Roscas probability of obtaining the pot is not related with ones immediate circumstances.
- In Bidding Roscas are not helpful - Deal with situations with cannot recur
- In LDCs risks are usually covariant• For Roscas to operate successfully individuals must commit
to pay into the Rosca after they win the pot.
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The Model
• n individuals• One indivisible durable consumption good• No access to credit markets• Each individual has T years of lifetime• Exogenous flow of income – y > 0• Identical intertemporally additive preferences• Instantaneous utilities depend on non-durable consumption
and whether or not the service of durable is enjoyed• Durable good doesn’t depreciate.• Cost of purchase of durable - B
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• Once purchased – the durable provides constant flow of services for rest of life
• Services of the durable are Non-fungible• No discounting – only motive to save is to buy the durable.• Instantaneous Utility,
- v(1,c) – owns the durable
- v(0,c) – without the durable• v’ > 0, v’’ < 0, and v(i,.) is three times continuously
differentiable.• Δv(c) v(1,c) – v(0,c)• v(α,c) αv(1,c) + (1-α)v(0,c) for 0 ≤ α ≤ 1• Δv(c) > 0, Δv’(c) ≥ 0
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Autarky• Individuals save up on their own.• It is optimal to save B at a constant rate y – c over an
interval [0,t]• Lifetime Utility under Autarky,
U(c,t) = t.v(0,c) + (T – t).v(1,y)• The individual’s objective is choosing t and c to
Maximize{t.v(0,c) + (T – t).v(1,y)} (1)
Subject to t(y – c) = B
and 0 ≤ c ≤ y
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• Let (ta,ca) be the solution to the maximization problem and Wa be
the maximal value of lifetime utility.
• To simplify the problem, substituting for t from (1), we get,
• Let
(2)
where 0 ≤ α ≤ 1
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• Setting α = 0 in (2), lifetime utility under autarky is
• The first term is lifetime utility if the durable were free.• The second term is the minimal utility cost of saving up for
the durable.• c*(α) is the consumption level which solves (2), the optimal
autarkic consumption rate, ca, is c*(0).
• No individual has the durable good before ta after which all n individuals receive it.
• Expected fraction of time that an individual will enjoy the services of the durable during accumulation period is 0. Hence, α is 0 in (2).
• Autarky is inefficient.
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LEMMA
Under the assumptions on preferences set out above,
• the minimized cost μ(.) in (2) is a decreasing, concave
functions of α
• the cost-minimizing consumption rate c*(.) is an increasing
function of α
• Both are twice continuously differentiable on [0,1], where
they satisfy the identity μ(α) v’(α,c*(α))
• Moreover, if Δv’’’(c)>0 for i= 0 and 1, and if Δv’’(c)≥0, then
c*(.) is strictly convex
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Random Roscas
• n-person group
• Meet at equally spaced dates up to ta, {ta /n, 2 ta /n,…, ta}.
• Each individual contributes B/n at each meeting
• At every meeting an individual is randomly selected to
receive the pot B, which allows him to buy the durable.
• The individual continues to save at rate B/ta over the interval
[0,ta]
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• Similar to autarky, but can now expect to receive the durable
ta(n+1)/2n
• The group’s objective is to maximize the (ex ante) expected
utility of the representative member.
• The choice variable here would be the length of the Rosca, t.
• Let the receipt date for the pot, τL be a random variable
distributed uniformly over the set {t/n, 2t/n,…,t}.
• Each member saves at rate B/t over the life of the Rosca.
• Non-durable consumption is thus c = y – B/t during this
period.
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• The agent’s problem is to maximize the lifetime utility:
τL.v(0,c)+(t- τL).v(1,c)+(T-t).v(1,y)
where t=B/(y-c)
• E(τL)=[(n+1)/2n]t. Substituting this in lifetime utility we get
each members (ex ante) welfare is
(4)
where t=B/(y-c)
• Group’s Problem: Choose t to maximize (4).
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• Let α (n – 1)/2n, we can write equation (4) as,
• Let the optimal length be tr , the associated consumption rate be cr and the maximal value of expected utility be Wr
• Analogous to the autarky case, we obtain the equation,
(5)
with cr = c*(α)
• From the lemma, we know μ(0) > μ(α) • Each member expects to enjoy the durable’s services for a
fraction α of the time while he is savings for the durable.
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PROPOSITION 1
• By forming a random Rosca, group members raise their
expected lifetime utilities.
• The optimal random Rosca involves members saving at a
lower rate over a longer interval than under autarky.
• Individuals expect to receive the durable good sooner in the
optimal random Rosca than under autarky
i.e. tr > ta > (n+1)tr/2n
under the restrictions v’’’(i,c) > 0 for i = 0,1 and Δv’’(c) ≥ 0
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Effects of Random Roscas
• Welfare is raised
• Reduction in everyone’s utility cost of saving up
• Non – durable consumption is higher
• Accumulation period is longer
• Individuals receive the good earlier on average
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Effects of Random Roscas (Ex-post)• Individuals have same prospects ex-ante• Circumstances differ once order of receipt has been
determined• Lets use index i to denote the person who wins the pot at the
ith meeting at date tr(i/n)
• Ex-post utilities are,
for i = 1,…,n
• For the final individual (cr , tr) is not optimal, he is strictly worse off
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Bidding Roscas
• Individuals bid for the right to receive the pot at a certain
date
• Order of receipt is determined at time 0
• Individuals have identical preferences. So they must end up
getting the same utility.
• Any efficient auction procedure should have the total
contributions in each period equal to B
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Setup of the Rosca
• Let the length of the Rosca be ‘t’
• At period 0, a series of n-1 ascending bid auctions are
conducted among n members to determine who receives the
pot at each auction
• The last individual has his contribution set such that the sum
of all contributions must equal the durable’s cost B.
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• Since the length of the Rosca is t, and there are n individuals
so the meeting times will be
{t/n, 2t/n,…, t}
• Let bi be the contribution of individual i, who wins the pot at
time (i/n)t.
• For a member bidding bi, his non-durable consumption would
be ci= y-(n/t)bi in each period
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• So the individual's utility is
• Here αi = (n-i)/n and since all individuals have the same utility,
say x, we have
v(αi, ci)= x, i=1,…,n
• Let cS be the average non durable consumption rate of members
during the life of rosca. Then we have
t(y- cT ) = B
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• Now given x we can, from the equations, find out ci’s and
Rosca length t
• From the earlier discussions we can write the welfare
functions as
T.v(1,y) - B {[ v ( 1 , y ) - x ] / [y -cT ]}
• Let c(α,x) be the function satisfying v(α,c) = x and define
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• Let tb and Wb be the optimal time and welfare level of the
bidding rosca.
• As before we can write
Wb = T. v(1,y) - B. μb (9)
Where (10)
• Let x* be the minimum in (10) we get, the length of the
optimal bidding rosca,
tb = B/[y- cT (x*)]
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PROPOSITION 2
• By forming a bidding Rosca, group members raise their
lifetime utility relative to autarky.
• Again if 1/v’(0,.) is concave, the optimal bidding Rosca
involves group members saving at a lower average rate and
over a longer interval than under autarky.
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Effect of Bidding Roscas• Welfare is raised
• Each individual has a different rate of non durable
consumption in the accumulation period
• Earlier acquirers of the durable bid a higher contribution to
the rosca and consume lesser of the non durable (c1<c2<…
<cn )
• Last individual to acquire the durable has greater non
durable consumption than under autarky
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Bidding versus Random Roscas• Roscas are in general more efficient than autarky• Bidding and Random Roscas lead to different outcomes
PROPOSITION 3
Group members’ expected utility will be higher if they use a random rather than a bidding Rosca. If the value of the durable is independent of the non-durable consumption rate [i.e., Δv’(c) 0], and if 1/v’(0,.) is a convex function, then the optimal random Rosca involves members saving at a lower rate over a longer interval than the optimal bidding Rosca.
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Intuition Behind Proposition 3• Looking from an ex-ante view point, consider two Roscas of
the same duration• Under Bidding – lifetime utilities are equal• Under Random Roscas – non-durable consumption rates are
the same.• Consider a General Scheme
- Order of receipt of pot is randomly assigned
- a consumption rate ci
- but the above constraints do not apply.• To maximize ex-ante expected welfare the scheme would
equate the marginal utilities
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• Random assignment with equal non-durable consumption is closer to this condition than bidding.
• In the bidding equilibrium all the members are on different points of the same indifference curve.
• The higher bidder has lesser consumption level so the marginal utility is higher than those receiving the pot later.
• In the Random allocation, the c and α are same ex-ante, so the marginal utilities of non-durable consumption are same.
• Thus, the Random Rosca performs better than the bidding Rosca
• When Δv’ 0 i.e. the marginal utility from having the durable is not affected by changes in the non-durable consumption then it reinforces the above fact.
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Proposition 3 Diagrammatically
• Group of 2 individuals
• Time horizon, T = 3 years
• Rosca length, t = 2 years
• Value of durable’s services, Δv(c) ξ > 0 is a constant i.e.
Δv’ 0
• Total annual consumption = 2y - B
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• In a Random Rosca the utility allocation is at point A or
point B because of equal non-durable consumption
• The slope of the relevant UPF is -1 at A and B
• Since Δv’ 0, the consumption at A and B are equal.
• Both A and B are equally probable so ex-ante expected
utility is at C
• In a Bidding Rosca, since the utilities have to be the same,
we finally end up at the intersection of the two frontiers
which is inferior to C.
• So a Random Rosca is better than a bidding Rosca
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Conclusion• We looked at an informal institution of Roscas
• They help individuals without access to credit markets to
pool in their resources and increase welfare.
• We looked at Random Roscas and Bidding Roscas
• In both cases individuals were better off participating in the
Roscas
• Comparing the two, with the assumption of identical
preferences, Random Roscas were better than Bidding
Roscas.